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Theorem cnmpt1vsca 18223
Description: Continuity of scalar multiplication; analogue of cnmpt12f 17698 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
tlmtrg.f  |-  F  =  (Scalar `  W )
cnmpt1vsca.t  |-  .x.  =  ( .s `  W )
cnmpt1vsca.j  |-  J  =  ( TopOpen `  W )
cnmpt1vsca.k  |-  K  =  ( TopOpen `  F )
cnmpt1vsca.w  |-  ( ph  ->  W  e. TopMod )
cnmpt1vsca.l  |-  ( ph  ->  L  e.  (TopOn `  X ) )
cnmpt1vsca.a  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )
cnmpt1vsca.b  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )
Assertion
Ref Expression
cnmpt1vsca  |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B
) )  e.  ( L  Cn  J ) )
Distinct variable groups:    x, F    x, J    x, K    x, L    ph, x    x, W    x, X
Allowed substitution hints:    A( x)    B( x)    .x. ( x)

Proof of Theorem cnmpt1vsca
StepHypRef Expression
1 cnmpt1vsca.l . . . . . . 7  |-  ( ph  ->  L  e.  (TopOn `  X ) )
2 cnmpt1vsca.w . . . . . . . . 9  |-  ( ph  ->  W  e. TopMod )
3 tlmtrg.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
43tlmscatps 18220 . . . . . . . . 9  |-  ( W  e. TopMod  ->  F  e.  TopSp )
52, 4syl 16 . . . . . . . 8  |-  ( ph  ->  F  e.  TopSp )
6 eqid 2436 . . . . . . . . 9  |-  ( Base `  F )  =  (
Base `  F )
7 cnmpt1vsca.k . . . . . . . . 9  |-  K  =  ( TopOpen `  F )
86, 7istps 17001 . . . . . . . 8  |-  ( F  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  F )
) )
95, 8sylib 189 . . . . . . 7  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  F )
) )
10 cnmpt1vsca.a . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )
11 cnf2 17313 . . . . . . 7  |-  ( ( L  e.  (TopOn `  X )  /\  K  e.  (TopOn `  ( Base `  F ) )  /\  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )  ->  ( x  e.  X  |->  A ) : X --> ( Base `  F ) )
121, 9, 10, 11syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> ( Base `  F )
)
13 eqid 2436 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1413fmpt 5890 . . . . . 6  |-  ( A. x  e.  X  A  e.  ( Base `  F
)  <->  ( x  e.  X  |->  A ) : X --> ( Base `  F
) )
1512, 14sylibr 204 . . . . 5  |-  ( ph  ->  A. x  e.  X  A  e.  ( Base `  F ) )
1615r19.21bi 2804 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  F
) )
17 tlmtps 18217 . . . . . . . . 9  |-  ( W  e. TopMod  ->  W  e.  TopSp )
182, 17syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  TopSp )
19 eqid 2436 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
20 cnmpt1vsca.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
2119, 20istps 17001 . . . . . . . 8  |-  ( W  e.  TopSp 
<->  J  e.  (TopOn `  ( Base `  W )
) )
2218, 21sylib 189 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  ( Base `  W )
) )
23 cnmpt1vsca.b . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )
24 cnf2 17313 . . . . . . 7  |-  ( ( L  e.  (TopOn `  X )  /\  J  e.  (TopOn `  ( Base `  W ) )  /\  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )  ->  ( x  e.  X  |->  B ) : X --> ( Base `  W ) )
251, 22, 23, 24syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  B ) : X --> ( Base `  W )
)
26 eqid 2436 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
2726fmpt 5890 . . . . . 6  |-  ( A. x  e.  X  B  e.  ( Base `  W
)  <->  ( x  e.  X  |->  B ) : X --> ( Base `  W
) )
2825, 27sylibr 204 . . . . 5  |-  ( ph  ->  A. x  e.  X  B  e.  ( Base `  W ) )
2928r19.21bi 2804 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  W
) )
30 eqid 2436 . . . . 5  |-  ( .s f `  W )  =  ( .s f `  W )
31 cnmpt1vsca.t . . . . 5  |-  .x.  =  ( .s `  W )
3219, 3, 6, 30, 31scafval 15969 . . . 4  |-  ( ( A  e.  ( Base `  F )  /\  B  e.  ( Base `  W
) )  ->  ( A ( .s f `  W ) B )  =  ( A  .x.  B ) )
3316, 29, 32syl2anc 643 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( A ( .s f `  W ) B )  =  ( A  .x.  B ) )
3433mpteq2dva 4295 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .s f `  W ) B ) )  =  ( x  e.  X  |->  ( A  .x.  B
) ) )
3530, 20, 3, 7vscacn 18215 . . . 4  |-  ( W  e. TopMod  ->  ( .s f `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
362, 35syl 16 . . 3  |-  ( ph  ->  ( .s f `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
371, 10, 23, 36cnmpt12f 17698 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .s f `  W ) B ) )  e.  ( L  Cn  J
) )
3834, 37eqeltrrd 2511 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B
) )  e.  ( L  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   TopOpenctopn 13649   .s fcscaf 15951  TopOnctopon 16959   TopSpctps 16961    Cn ccn 17288    tX ctx 17592  TopModctlm 18187
This theorem is referenced by:  tlmtgp  18225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-slot 13473  df-base 13474  df-topgen 13667  df-scaf 15953  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cn 17291  df-tx 17594  df-tmd 18102  df-tgp 18103  df-trg 18189  df-tlm 18191
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